Primitive Words, Free Factors and Measure Preservation

Let F_k be the free group on k generators. A word w \in F_k is called primitive if it belongs to some basis of F_k. We investigate two criteria for primitivity, and consider more generally, subgroups of F_k which are free factors. The first criterion is graph-theoretic and uses Stallings core graphs: given subgroups of finite rank H \le J \le F_k we present a simple procedure to determine whether H is a free factor of J. This yields, in particular, a procedure to determine whether a given element in F_k is primitive. Again let w \in F_k and consider the word map w:G x G x ... x G \to G (from the direct product of k copies of G to G), where G is an arbitrary finite group. We call w measure preserving if given uniform measure on G x G x ... x G, w induces uniform measure on G (for every finite G). This is the second criterion we investigate: it is not hard to see that primitivity implies measure preservation and it was conjectured that the two properties are equivalent. Our combinatorial approach to primitivity allows us to make progress on this problem and in particular prove the conjecture for k=2. It was asked whether the primitive elements of F_k form a closed set in the profinite topology of free groups. Our results provide a positive answer for F_2.Comment: This is a unified version of two manuscripts: "On Primitive words I: A New Algorithm", and "On Primitive Words II: Measure Preservation". 42 pages, 14 figures. Some parts of the paper reorganized towards publication in the Israel J. of Mat

Cosmic censorship of smooth structures

It is observed that on many 4-manifolds there is a unique smooth structure underlying a globally hyperbolic Lorentz metric. For instance, every contractible smooth 4-manifold admitting a globally hyperbolic Lorentz metric is diffeomorphic to the standard $\R^4$. Similarly, a smooth 4-manifold homeomorphic to the product of a closed oriented 3-manifold $N$ and $\R$ and admitting a globally hyperbolic Lorentz metric is in fact diffeomorphic to $N\times \R$. Thus one may speak of a censorship imposed by the global hyperbolicty assumption on the possible smooth structures on $(3+1)$-dimensional spacetimes.Comment: 5 pages; V.2 - title changed, minor edits, references adde

Shadows and traces in bicategories

Traces in symmetric monoidal categories are well-known and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some applications, such as generalizations of the Lefschetz theorem, one needs "noncommutative" traces, such as the Hattori-Stallings trace for modules over noncommutative rings. In this paper we study a generalization of the symmetric monoidal trace which applies to noncommutative situations; its context is a bicategory equipped with an extra structure called a "shadow." In particular, we prove its functoriality and 2-functoriality, which are essential to its applications in fixed-point theory. Throughout we make use of an appropriate "cylindrical" type of string diagram, which we justify formally in an appendix.Comment: 46 pages; v2: reorganized and shortened, added proof for cylindrical string diagrams; v3: final version, to appear in JHR

Genetic factors are important determinants of impaired growth after infant cardiac surgery

ObjectivesWe sought to estimate the prevalence and identify the predictors of impaired growth after infant cardiac surgery.MethodsWe performed a secondary analysis of a prospective study of the role of apolipoprotein E gene polymorphisms on neurodevelopment in young children after infant cardiac surgery. Prevalence estimates for growth velocity were derived by using anthropometric measures (weight and head circumference) obtained at birth and at 4 years of age. Genetic evaluation was also performed. Growth measure z scores were calculated by using World Health Organization Child Growth Standards. Growth velocity was evaluated by using 2 different techniques: first by clustering the children into one of 3 growth velocity subgroups based on z scores (impaired growth, difference < −0.5 standard deviation; stable growth, difference of −0.5 to 0.5 standard deviation; and improving growth, difference > 0.5 SD) and second by using continuous difference scores. Statistical analyses were conducted with a combination of proportional odds models for the ordered categories and simple linear regression for the continuous outcomes.ResultsThree hundred nineteen full-term subjects had complete anthropometric measures for weight and head circumference at birth and 4 years. The cohort was 56% male. Genetic examinations were available for 97% (309/319) of the cohort (normal, 74%; definite or suspected genetic abnormality, 26%). Frequency counts for weight categories were as follows: impaired growth, 37%; stable growth, 31%; and improving growth, 32%. Frequency counts for head circumference categories were as follows: impaired growth, 39%; stable growth, 28%; and improving growth, 33%. The presence of a definite or suspected genetic syndrome (P = .04) was found to be a predictor of impaired growth for weight but not for head circumference. When growth z scores were used as continuous outcomes, the apolipoprotein E ε2 allele was found to be predictive of lower z scores for both weight (P = .02) and head circumference (P = .03).ConclusionsImpaired growth for both weight and head circumference is common (both >30%) in this cohort of children after infant cardiac surgery. Both the apolipoprotein E ε2 allele and the presence of a definite or suspected genetic syndrome were associated with impaired weight growth velocity. The apolipoprotein E ε2 allele was also associated with impaired growth velocity for head circumference. Persistent poor growth might have long-term implications for the health and development of children with congenital heart defects

Free subgroups of one-relator relative presentations

Suppose that G is a nontrivial torsion-free group and w is a word over the alphabet G\cup\{x_1^{\pm1},...,x_n^{\pm1}\}. It is proved that for n\ge2 the group \~G= always contains a nonabelian free subgroup. For n=1 the question about the existence of nonabelian free subgroups in \~G is answered completely in the unimodular case (i.e., when the exponent sum of x_1 in w is one). Some generalisations of these results are discussed.Comment: V3: A small correction in the last phrase of the proof of Theorem 1. 4 page

Effects of preservation methods of muscle tissue from upper-trophic level reef fishes on stable isotope values (δ13C and δ15N)

© The Author(s), 2015. This article is distributed under the terms of the Creative Commons Attribution License. The definitive version was published in PeerJ 3 (2015): e874, doi:10.7717/peerj.874.Research that uses stable isotope analysis often involves a delay between sample collection in the field and laboratory processing, therefore requiring preservation to prevent or reduce tissue degradation and associated isotopic compositions. Although there is a growing literature describing the effects of various preservation techniques, the results are often contextual, unpredictable and vary among taxa, suggesting the need to treat each species individually. We conducted a controlled experiment to test the effects of four preservation methods of muscle tissue from four species of upper trophic-level reef fish collected from the eastern Gulf of Mexico (Red Grouper Epinephelus morio, Gag Mycteroperca microlepis, Scamp Mycteroperca phenax, and Red Snapper Lutjanus campechanus). We used a paired design to measure the effects on isotopic values for carbon and nitrogen after storage using ice, 95% ethanol, and sodium chloride (table salt), against that in a liquid nitrogen control. Mean offsets for both δ13C and δ15N values from controls were lowest for samples preserved on ice, intermediate for those preserved with salt, and highest with ethanol. Within species, both salt and ethanol significantly enriched the δ15N values in nearly all comparisons. Ethanol also had strong effects on the δ13C values in all three groupers. Conversely, for samples preserved on ice, we did not detect a significant offset in either isotopic ratio for any of the focal species. Previous studies have addressed preservation-induced offsets in isotope values using a mass balance correction that accounts for changes in the isotope value to that in the C/N ratio. We tested the application of standard mass balance corrections for isotope values that were significantly affected by the preservation methods and found generally poor agreement between corrected and control values. The poor performance by the correction may have been due to preferential loss of lighter isotopes and corresponding low levels of mass loss with a substantial change in the isotope value of the sample. Regardless of mechanism, it was evident that accounting for offsets caused by different preservation methods was not possible using the standard correction. Caution is warranted when interpreting the results from specimens stored in either ethanol or salt, especially when using those from multiple preservation techniques. We suggest the use of ice as the preferred preservation technique for muscle tissue when conducting stable isotope analysis as it is widely available, inexpensive, easy to transport and did not impart a significant offset in measured isotopic values. Our results provide additional evidence that preservation effects on stable isotope analysis can be highly contextual, thus requiring their effects to be measured and understood for each species and isotopic ratio of interest before addressing research questions.Funding was provided by a grant to CD Stallings and TS Switzer from the National Oceanic and Atmospheric Administration, Cooperative Research Program (NA12NMF4540081)

3-manifolds which are spacelike slices of flat spacetimes

We continue work initiated in a 1990 preprint of Mess giving a geometric parameterization of the moduli space of classical solutions to Einstein's equations in 2+1 dimensions with cosmological constant 0 or -1 (the case +1 has been worked out in the interim by the present author). In this paper we make a first step toward the 3+1-dimensional case by determining exactly which closed 3-manifolds M^3 arise as spacelike slices of flat spacetimes, and by finding all possible holonomy homomorphisms pi_1(M^3) to ISO(3,1).Comment: 10 page

Socioeconomic differentials in the immediate mortality effects of the national Irish smoking ban

This article has been made available through the Brunel Open Access Publishing Fund.Background: Consistent evidence has demonstrated that smoking ban policies save lives, but impacts on health inequalities are uncertain as few studies have assessed post-ban effects by socioeconomic status (SES) and findings have been inconsistent. The aim of this study was to assess the effects of the national Irish smoking ban on ischemic heart disease (IHD), stroke, and chronic obstructive pulmonary disease (COPD) mortality by discrete and composite SES indicators to determine impacts on inequalities. Methods: Census data were used to assign frequencies of structural and material SES indicators to 34 local authorities across Ireland with a 2000–2010 study period. Discrete indicators were jointly analysed through principal component analysis to generate a composite index, with sensitivity analyses conducted by varying the included indicators. Poisson regression with interrupted time-series analysis was conducted to examine monthly age and gender-standardised mortality rates in the Irish population, ages ≥35 years, stratified by tertiles of SES indicators. All models were adjusted for time trend, season, influenza, and smoking prevalence. Results: Post-ban mortality reductions by structural SES indicators were concentrated in the most deprived tertile for all causes of death, while reductions by material SES indicators were more equitable across SES tertiles. The composite indices mirrored the results of the discrete indicators, demonstrating that post-ban mortality decreases were either greater or similar in the most deprived when compared to the least deprived for all causes of death. Conclusions: Overall findings indicated that the national Irish smoking ban reduced inequalities in smoking-related mortality. Due to the higher rates of smoking-related mortality in the most deprived group, even equitable reductions across SES tertiles resulted in decreases in inequalities. The choice of SES indicator was influential in the measurement of effects, underscoring that a differentiated analytical approach aided in understanding the complexities in which structural and material factors influence mortality

Reductions in cardiovascular, cerebrovascular, and respiratory mortality following the national Irish smoking ban: Interrupted time-series analysis

Copyright @ 2013 Stallings-Smith et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.This article has been made available through the Brunel Open Access Publishing Fund.Background: Previous studies have shown decreases in cardiovascular mortality following the implementation of comprehensive smoking bans. It is not known whether cerebrovascular or respiratory mortality decreases post-ban. On March 29, 2004, the Republic of Ireland became the first country in the world to implement a national workplace smoking ban. The aim of this study was to assess the effect of this policy on all-cause and cause-specific, non-trauma mortality. Methods: A time-series epidemiologic assessment was conducted, utilizing Poisson regression to examine weekly age and gender-standardized rates for 215,878 non-trauma deaths in the Irish population, ages ≥35 years. The study period was from January 1, 2000, to December 31, 2007, with a post-ban follow-up of 3.75 years. All models were adjusted for time trend, season, influenza, and smoking prevalence. Results: Following ban implementation, an immediate 13% decrease in all-cause mortality (RR: 0.87; 95% CI: 0.76-0.99), a 26% reduction in ischemic heart disease (IHD) (RR: 0.74; 95% CI: 0.63-0.88), a 32% reduction in stroke (RR: 0.68; 95% CI: 0.54-0.85), and a 38% reduction in chronic obstructive pulmonary disease (COPD) (RR: 0.62; 95% CI: 0.46-0.83) mortality was observed. Post-ban reductions in IHD, stroke, and COPD mortalities were seen in ages ≥65 years, but not in ages 35-64 years. COPD mortality reductions were found only in females (RR: 0.47; 95% CI: 0.32-0.70). Post-ban annual trend reductions were not detected for any smoking-related causes of death. Unadjusted estimates indicate that 3,726 (95% CI: 2,305-4,629) smoking-related deaths were likely prevented post-ban. Mortality decreases were primarily due to reductions in passive smoking. Conclusions: The national Irish smoking ban was associated with immediate reductions in early mortality. Importantly, post-ban risk differences did not change with a longer follow-up period. This study corroborates previous evidence for cardiovascular causes, and is the first to demonstrate reductions in cerebrovascular and respiratory causes